A NORMAL MEASURE ON A COMPACT CONNECTED SPACE
GRZEGORZ PLEBANEK
Abstract. We present a construction of a compact connected space which supports a normal probability measure.
1. Introduction
If K is a compact Hausdorff space then we denote by P (K) the set of all probability regular Borel measures on K. We write Z(K) for the family of all closed Gδ subsets of K. Since every compact space is normal, Z ∈ Z(K) if and only if Z is a zero set, i.e.
Z = f−1(0) for some continuous function f : K → R.
A measure µ ∈ P (K) is normal if µ is order-continuous on the Banach lattice C(K).
Equivalently, µ(F ) = 0 whenever F ⊆ K is a closed set with empty interior ([1], Theorem 4.6.3). A typical example of a normal measure is the natural measure defined on the Stone space of the measure algebra A of the Lebesgue measure λ on [0, 1]. Since the algebra A is complete, its Stone space is extremely disconnected.
By a result from [2] if K is a locally connected compactum then no measure µ ∈ P (K) can be normal, cf. [1], Proposition 4.6.20. Dales et al. posed a problem that can be stated as follows (Question 2 in [1]).
Problem 1.1. Suppose that K is a compact and µ ∈ P (K) is a normal measure. Must K be disconnected?
We show below that the answer is negative, namely we prove the following result.
Theorem 1.2. There is a compact connected space L of weight c which is the support of a normal measure.
2. Preliminaries
Recall that µ ∈ P (K) is said to be strictly positive or fully supported by K if µ(U ) > 0 for every non-empty open set U ⊆ K.
Lemma 2.1. Let K be a compact space, and suppose that µ is a strictly positive measure on K such that µ(Z) = 0 for every Z ∈ Z(K) with empty interior. Then µ is a normal measure.
July 5, 2014. I wish to thank H. Garth Dales for the discussion concerning the subject of this note.
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Proof. Assume that there is a closed set F ⊆ K with empty interior but with µ(F ) > 0.
Then we derive a contradiction by the following observation.
Claim. Every closed set F ⊆ K with empty interior is contained in some Z ∈ Z(K) with empty interior.
Indeed, consider a maximal family F of continuous functions K → [0, 1] such that f |F = 0 for f ∈ F and f · g = 0 whenever f, g ∈ F , f 6= g. Then F is necessarily countable because K, being the support of a measure, satisfies the countable chain condition. Write F = {fn : n ∈ N} and let f = P
n2−nfn and Z = f−1(0). Then the function f is continuous so that Z ∈ Z(K). We have Z ⊇ F and the interior of Z must
be empty by the maximality of F .
If f : K → L is a continuous map and µ ∈ P (K) then the measure f [µ] ∈ P (L) is defined by f [µ](B) = µ(f−1(B)) for every Borel set B ⊆ L.
We shall consider inverse systems of compact spaces with measures of the form hKα, µα, παβ : β < α < κi,
where κ is an ordinal number and for all γ < β < α < κ we have 2(i) Kα is a compact space and µα ∈ P (Kα);
2(ii) παβ : Kα → Kβ is a continuous surjection;
2(iii) πβγ ◦ παβ = παγ; 2(iv) παβ[µα] = µβ.
The following summarises basic facts on inverse systems satisfying 2(i)-(iv).
Theorem 2.2. Let K be the limit of the system with uniquely defined continuous sur- jections πα : K → Kα for α < κ.
(a) K is a compact space and K is connected whenever all the space Kα are connected.
(b) There is the unique µ ∈ P (K) such that πα[µ] = µα for α < κ.
(c) If every µα is strictly positive then µ is strictly positive.
Engelking’s General Topology contains the topological part of 2.2 (measure-theoretic ingredients call for a proper reference). We also use the following fact on closed Gδ sets and inverse systems of length ω1.
Lemma 2.3. Let K be the limit of an inverse system hKα, πβα : β < α < ω1i. Then for every Z ∈ Z(K), there are α < ω1 and Zα ∈ Z(Kα) with Z = πα−1(Zα).
Proof. Sets of the form π−1α (V ), where α < κ and V ⊆ Kα is open, give the canonical basis of K (closed under countable unions). Therefore if Z ∈ Z(K) then Z =T
nπ−1αn(Vn) for some αn < ω1 and some open Vn ⊆ Kαn. Taking α > supnαn we can write Z = T
nπα−1(Wn) for some open Wn⊆ Kα. Let Zα =T
nWn. Then Zα is Gδ in Kα, π−1α (Zα) =
Z and Zα = πα(Z) is closed.
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3. Proof of Theorem 1.2
We first describe a basic construction which will be used repeatedly.
Lemma 3.1. Let K be a compact connected space, and let µ ∈ P (K) be a strictly positive measure. If F ⊆ K is a closed set with µ(F ) > 0, then there are a compact connected space bK, a strictly positive measure bµ ∈ P ( bK) and a continuous surjection f : bK → K such that f [µ] = µ and int((fb −1(F )) 6= ∅.
Proof. Let F0 be the support of µ restricted to F , that is F0 = F \[
{U : U open and µ(F ∩ U ) = 0}.
Let bK = {(x, t) ∈ K × [0, 1] : x ∈ F0 or t = 0}. Then bK is clearly a compact connected space and f (x, t) = x defines a continuous surjection f : bK → K. Moreover, the set f−1(F ) contains F0× [0, 1], a set with non-empty interior. Hence int(f−1(F )) 6= ∅.
We can define µ ∈ P ( bb K) with the required property by setting bµ(B) = µ(f (B ∩ (K \ F ) × {0})) + µ ⊗ λ(F × [0, 1] ∩ B),
for Borel sets B ⊆ bK, where λ is the Lebesgue measure on [0, 1]. Lemma 3.2. Let K be a compact connected space, and let µ ∈ P (K) be a strictly positive measure. Then there are a compact connected space K#, a strictly positive measure µ#∈ P (K#) and a continuous surjection g : K#→ K such that g[µ#] = µ and int((g−1(Z)) 6= ∅ for every Z ∈ Z(K) with µ(Z) > 0.
Proof. Let {Zα : α < κ} be an enumeration of all sets Z ∈ Z(K) of positive measure.
Setting K0 = K, µ0 = µ we define inductively an inverse system hKα, µα, παβ : β < α < κi satisfying 2(i)-(iv). Assume the construction for all α < ξ.
If ξ is the limit ordinal we use Theorem 2.2 and let Kξ be the limit of Kα, α < κ, and µξ be the unique measure as in 2.3.
If ξ = α + 1 then we define Kξ and µξ ∈ P (Kξ) applying Lemma 3.1 to K = Kα, µ = µα, F = (πα0)−1(Zα).
Then we can define K# and µ# as the limit of hKα, µα, πβα : β ≤ α < κi and set g = π0 : K#→ K.
Indeed, if Z ∈ Z(K) and µ(Z) > 0 then Z = Zα for some α < κ so the interior of the set
(π0α+1)−1(Zα) = (παα+1)−1((π0α)−1(Zα),
is nonempty by the basic construction of Lemma 3.1. It follows that int(g−1(Zα)) 6= ∅,
and we are done.
We are now ready for the proof of Theorem 1.2. Let L0 = [0, 1] and µ0 = λ. Using Lemma 3.2 we define an inverse system hLα, µα, πβα : β ≤ α < ω1i, where Lα+1 = (Lα)#
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and µα+1 = (µα)#. Consider the limit L of this inverse system with the limit measure ν ∈ P (L).
We shall check that ν is a normal measure using Lemma 2.1. Take Z ∈ Z(L) with ν(Z) > 0. It follows from Lemma 2.3 that Z = πα−1(Zα) for some α < ω1 and Zα ∈ Z(Lα). Then the set (πα+1α )−1(Zα) has non-empty interior in Lα+1 = (Lα)# and, consequently, int(Z) 6= ∅.
Note that in a compact space K of topological weight w(K) ≤ c there are at most c many closed Gδ sets. It follows from the proof of Lemma 3.2 that w(K#) ≤ c whenever w(K) ≤ c. Therefore w(Lα) ≤ c for every α < ω1 and w(L) = c. This finishes the proof of our main result.
Let us remark that using Lemma 3.1 and the construction from Kunen [3] one can prove the following variant of Theorem 1.2.
Theorem 3.3. Assuming the continuum hypothesis, there is a perfectly normal compact connected space L supporting a normal probability measure.
Perfect normality of L means that every closed subset of L is Gδ so in particular the space L from Theorem 3.3 is first-countable.
References
[1] H.G. Dales, F.K. Dashiell Jr., A. T.-M. Lau, D. Strauss Banach Spaces of Continuous Functions as Dual Spaces, preprint (2014).
[2] B. Fishel, D. Papert, A Note on Hyperdiffuse Measures, J. London Math. Soc. s1-39 (1), (1964), 245-254.
[3] K. Kunen, A compact L-space under CH, Topology Appl. 12 (1981), 283-287.
Instytut Matematyczny, Uniwersytet Wroc lawski E-mail address: grzes@math.uni.wroc.pl
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